Theorem reseq1d 4670

Description: Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)

Hypothesis

Ref	Expression
reseqd.1	|- ( ph -> A = B )

Assertion

Ref	Expression
reseq1d	|- ( ph -> ( A |` C ) = ( B |` C ) )

Proof of Theorem reseq1d

Step	Hyp	Ref	Expression
1		reseqd.1	. 2 |- ( ph -> A = B )
2		reseq1 4665	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A |` C ) = ( B |` C ) )

Syntax hints:   -> wi 4   = wceq 1285   |` cres 4403

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-in 2990  df-res 4413

This theorem is referenced by:  reseq12d  4672  fun2ssres  5010  funcnvres2  5042  funimaexg  5051  fresin  5137  offres  5841  tfrlemisucaccv  6022  tfrlemi1  6029  tfr1onlemsucaccv  6038  tfrcllemsucaccv  6051  freceq1  6089  freceq2  6090  fseq1p1m1  9401